We present a design for the computer systems which control, configure, and monitor the Atacama Large Millimeter Array (ALMA) correlator and process its output. Two distinct computer systems implement this functionality: a rack- mounted PC controls and monitors the correlator, and a cluster of 17 PCs process the correlator output into raw spectral results. The correlator computer systems interface to other ALMA computers via gigabit Ethernet networks utilizing CORBA and raw socket connections. ALMA Common Software provides the software infrastructure for this distributed computer environment. The control computer interfaces to the correlator via multiple CAN busses and the data processing computer cluster interfaces to the correlator via sixteen dedicated high speed data ports. An independent array-wide hardware timing bus connects to the computer systems and the correlator hardware ensuring synchronous behavior and imposing hard deadlines on the control and data processor computers. An aggregate correlator output of 1 gigabyte per second with 16 millisecond periods and computational data rates of approximately 1 billion floating point operations per second define other hard deadlines for the data processing computer cluster.